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Solution - Absolute value equations

Exact form:

v = 0, 0

v=0 , 0

See steps

Step-by-step explanation

1. Rewrite the equation without absolute value bars

Use the rules:
$| x | = | y |$ → $x = \pm y$ and $| x | = | y |$ → $\pm x = y$
to write all four options of the equation
$| 3 v | = | 3 v |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 3 v \| = \| 3 v \|$
$x = + y$	$(3 v) = (3 v)$
$x = - y$	$(3 v) = - (3 v)$
$+ x = y$	$(3 v) = (3 v)$
$- x = y$	$- (3 v) = (3 v)$

When simplified, equations $x = + y$ and $+ x = y$ are the same and equations $x = - y$ and $- x = y$ are the same, so we end up with only 2 equations:

$\| x \| = \| y \|$	$\| 3 v \| = \| 3 v \|$
$x = + y$ , $+ x = y$	$(3 v) = (3 v)$
$x = - y$ , $- x = y$	$(3 v) = - (3 v)$

2. Solve the two equations for v

2 additional steps

$3 v = 3 v$

Subtract from both sides:

$(3 v) - 3 v = (3 v) - 3 v$

Simplify the arithmetic:

$0 = (3 v) - 3 v$

Simplify the arithmetic:

$0 = 0$

6 additional steps

$3 v = - 3 v$

Divide both sides by :

$\frac{(3 v)}{3} = \frac{(- 3 v)}{3}$

Simplify the fraction:

$v = \frac{(- 3 v)}{3}$

Simplify the fraction:

$v = - v$

Add to both sides:

$v + v = - v + v$

Simplify the arithmetic:

$2 v = - v + v$

Simplify the arithmetic:

$2 v = 0$

Divide both sides by the coefficient:

$v = 0$

3. List the solutions

$v = 0, 0$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 3 v |$
$y = | 3 v |$
The equation is true where the two lines cross.

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Why learn this

We encounter absolute values almost daily. For example: If you walk 3 miles to school, do you also walk minus 3 miles when you go back home? The answer is no because distances use absolute value. The absolute value of the distance between home and school is 3 miles, there or back.
In short, absolute values help us deal with concepts like distance, ranges of possible values, and deviation from a set value.

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for v

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for v

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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